Circle segment

A quarter circle with a radius of 4 has the same area as a circle segment with a radius of 3. What is the magnitude of the center angle of the circle segment?

Correct answer:

α =  160 °

Step-by-step explanation:

$$\begin{gathered} r=4 \\ {S}_1=\frac{\pi \cdot r^2}{4}=\frac{3.1416\cdot 4^2}{4}\doteq 12.5664 \\ R=3 \\ {S}_2=\pi R^2\cdot \frac{\alpha }{360} \\ {S}_1=S_2 \\ \alpha =\frac{360\cdot S_1}{\pi \cdot R^2}=\frac{360\cdot 12.5664}{3.1416\cdot 3^2}=160=160^{\circ } \\ \text{ Verifying Solution: } \\ {S}_3=\pi \cdot R^2\cdot \frac{\alpha }{360}=3.1416\cdot 3^2\cdot \frac{160}{360}\doteq 12.5664 \end{gathered}$$

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